Discrete Mean Estimates and the Landau-Siegel Zero: Evaluation of Ξ11

2 Jun 2024


(1) Yitang Zhang.

  1. Abstract & Introduction
  2. Notation and outline of the proof
  3. The set Ψ1
  4. Zeros of L(s, ψ)L(s, χψ) in Ω
  5. Some analytic lemmas
  6. Approximate formula for L(s, ψ)
  7. Mean value formula I
  8. Evaluation of Ξ11
  9. Evaluation of Ξ12
  10. Proof of Proposition 2.4
  11. Proof of Proposition 2.6
  12. Evaluation of Ξ15
  13. Approximation to Ξ14
  14. Mean value formula II
  15. Evaluation of Φ1
  16. Evaluation of Φ2
  17. Evaluation of Φ3
  18. Proof of Proposition 2.5

Appendix A. Some Euler products

Appendix B. Some arithmetic sums


8. Evaluation of Ξ11

We first prove a general result as follows.

By Proposition 7.1, our goal is reduced to evaluating the sum


so that

Lemma 8.2. Suppose T < x < P. Then for µ = 6, 7


Proof. The sum is equal to

We move the contour of integration to the vertical segments

and to the two connecting horizontal segments

It follows by Lemma 5.6 that

The result now follows by direct calculation.

Combining these results with Lemma 8.3, we find that the integral (8.9) is equal to

The result now follows by direct calculation.

This paper is available on arxiv under CC 4.0 license.